## Functions in 2D and 3D

Functions of the form F(x) and F(x,y) containing up to 20 subfunctions, 30 special functions, 150 numerical and 100 named constants.

Functions of the form F(x) can be integrated, differentiated or searched for zeroes and extrema. Cartesian as well as Polar coordinates can be chosen and functions can be represented in parametrized form (i.e. y(t) versus x(t)).

F(x,y) can be represented by Shaded surfaces, Contourplots and Cross-sections through Contourplots. The 3D viewer provides a quick view of the shaded surface from different viewing angles and distances. Rotate the surface, zoom in/out or change the position of the light source by moving the mouse.

## Special Functions (real numbers only!)

Random numbers
Random(a)

Random(a) Generates a series of random numbers
between 0 and 1 from seed a. Below an example is given
F1=Random(0.5) calculated
for 100 x-values (F is not a function of x here)

Random steps with probability x: Ranstep(a,x)

Generates a random series of 0’s and 1’s from seed a.
The average fraction (or probablity) of 1’s is given by x.
Below an example is given of an Iteration (100 iterations):
F1=F2
F2=F1+ Ranstep(0.5,0.3)

Modulo function Mod(a,x)

Mod(a,x) = x – k.a, where integer k is chosen such that
0 < Mod(a,x) < a for a > 0 and
a < Mod(a,x) < 0 for a < 0

Fresnel Integrals
Sine and Cosine Fresnel Integral: C(x), S(x)

Exponential integrals and related functions
Exponential integral Ei(x)

(Scaled) Exponential integral En(n,x)

Note the exponential integral is often defined without the scaling factor exp(x)

Logarithmic integral Li(x)
Sine integral S(x)
Cosine integral C(x)
Hyperbolic sine integral Shi(x)
Hyperbolic cosine integral Chi(x)
Gamma function and related functions
Gamma function Gamma(x)

Wikipedia

Mathworld

Incomplete (lower) gamma function of the first kind Gammi(a,x)

Mathgrapher calculates the normalized function:

Complementary (upper) incomplete gamma function of the second kind Gammi(a,x)

Wikipedia

Riemann Zeta function Zeta(x)

This series is convergent for all numbers s>1. This holds for all complex numbers with real part >1. Euler showed that it is connected with the prime numbers
by the relation:

where the product runs over all prime numbers p.

Riemann found an analytical continuation to this function
for all s except 1

The integral runs from infinity to zero just above the real
axis, turns aorund zero and runs back to infinity just below the real axis
Mathgrapher calculates this function along the real axis (s=1 excluded)

The famous Riemann hypothesis states that all (non-trivial) zeroes of this function are on the line Re(s)=1/2. This Riemann hypothesis is unproved although more than
10^13 zeroes have been found, all of which ly on this line.

Bessel functions
Bessel functions of the first kind Bessel J(a,x)

Bessel functions of the second kind Bessel Y(a,x)

Elliptic integrals
Elliptic integral of the first kind K(x)

Elliptic integral of the second kind E(x)

Airy functions AI(x), BI(x)

The Airy functions AI(x) and BI(x) are the two linearly independant solutions of the second-order differential equation: y”-xy=0. For real values these functions are equal to the following integrals:

and

Probability function
Poisson probability function (or probability distribution function) Ppf(a,x)

Example:

The distribution gives the probability for m radioactive
decays in some fixed time interval, where a is the mean
number of events in that interval. For example: a = np, the product of the the number of radioactive nuclei in a source and the probability for a nucleus to decay in a that interval of time.

Binomial probability distribution function Bpf(a,b,x)

Gives the probability of m successes in n trials when the chance of success is p. In Mathgrapher this fuction is Bpf(a,b,x) where a=n, b=p, x=m

Wikipedia

Distribution functions
Binomial (cumulative) distribution function Bin(a,b,x)

Gives the probability that m or fewer successes occur in n independent trials when the chance of success is p. In Mathgrapher this is Bin(a,b,x) where a=n (integer), b=p and x=m (integer).

Poisson (cumulative) distribution function Pdf(a,x)

Gives the probability of m or less events in some fixed time interval, where a is the mean rate of events.

Wikipedia

Error Function Erf(x)

Wikipedia

Gaussian (standard normal) distribution function Gausdf(x)

A random distribution is often described by the Gauss probability distribution function

Mathgrapher gives the standard cumulative Gauss distribution function

Below some examples are given for this function Gausdf(x)

Wikipedia

The cumulative probability of the Beta distribution

Wikipedia

Chi squared distribution function (cumulative) Chidf(a,x)

The denumerator contains the Gamma function. In Mathgrapher this is Chidf(a,x) where a is the number of
degrees of freedom (a= 0.5 or larger).